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Multiple Linear and 1D Regression

多元线性回归

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Regression is the study of the conditional distribution Y |x of the response Y given the p × 1 vector of nontrivial predictors x. In a 1D regression model, Y is conditionally independent of x given a single linear combination α + βTx of the predictors, written Y x|(α + βTx) or Y x|βTx. Many of the most used statistical methods are 1D models, including generalized linear models such as multiple linear regression, logistic regression, and Poisson regression

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Multiple Linear and 1D Regression

David J. Olive

Southern Illinois University

Department of Mathematics

Mailcode 4408

Carbondale, IL 62901-4408

dolive@math.siu.edu

January 4, 2010

Contents

Preface vi

1 Introduction 1

1.1 Multiple Linear Regression ................. 5

1.2 Logistic Regression ...................... 9

1.3 Poisson Regression ....................... 12

1.4 Single Index Models ...................... 16

1.5 Survival Regression Models ................. 19

1.6 Variable Selection ....................... 20

1.7 Other Issues ........................... 25

1.8 Complements .......................... 27

1.9 Problems ............................. 27

2 Multiple Linear Regression 28

2.1 The MLR Model ........................ 28

2.2 Checking Goodness of Fit .................. 31

2.3 Checking Lack of Fit ..................... 35

2.3.1 Residual Plots ..................... 36

2.3.2 Other Model Violations ............... 40

2.4 The ANOVA F TEST ..................... 42

2.5 Prediction ............................ 49

2.6 The Partial F or Change in SS TEST ........... 56

2.7 The Wald t Test ........................ 61

2.8 The OLS Criterion ....................... 64

2.9 Two Important Special Cases ................ 68

2.9.1 The Location Model .................. 68

2.9.2 Simple Linear Regression .............. 69

2.10 The No Intercept MLR Model ............... 71

2.11 Summary ............................. 74

2.12 Complements .......................... 77

2.12.1 Lack of Fit Tests .................... 79

2.13 Problems ............................. 81

3 Building an MLR Model 102

3.1 Predictor Transformations ..................103

3.2 Graphical Methods for Response Transformations . . . 109

3.3 Main Eﬀects, Interactions and Indicators ........116

3.4 Variable Selection .......................118

3.5 Diagnostics ...........................141

3.6 Outlier Detection .......................146

3.7 Summary .............................151

3.8 Complements ..........................155

3.9 Problems .............................160

4 WLS and Generalized Least Squares 181

4.1 Random Vectors ........................181

4.2 GLS, WLS and FGLS .....................183

4.3 Inference for GLS .......................188

4.4 Complements ..........................191

4.5 Problems .............................191

5 One Way ANOVA 194

5.1 Introduction ...........................194

5.2 Fixed Eﬀects One Way ANOVA ..............196

5.3 Random Eﬀects One Way ANOVA ............207

5.4 Response Transformations for Experimental Design . . 209

5.5 Summary .............................211

5.6 Complements ..........................216

5.7 Problems .............................222

6 K Way ANOVA 234

6.1 Two Way ANOVA .......................234

6.2 kWayAnovaModels .....................240

6.3 Summary .............................240

6.4 Complements ..........................243

6.5 Problems .............................243

7 Block Designs 248

7.1 One Way Block Designs ...................249

7.2 Blocking with the K Way Anova Design .........253

7.3 Latin Square Designs .....................255

7.4 Summary .............................259

7.5 Complements ..........................262

7.6 Problems .............................263

8 Orthogonal Designs 267

8.1 Factorial Designs ........................267

8.2 Fractional Factorial Designs .................283

8.3 Plackett Burman Designs ..................288

8.4 Summary .............................291

8.5 Complements ..........................303

8.6 Problems .............................304

9 More on Experimental Designs 311

9.1 Split Plot Designs .......................311

9.1.1 Whole Plots Randomly Assigned to A ......312

9.1.2 Whole Plots Assigned to A as in a CRBD ....314

9.2 Review of the DOE Models .................317

9.3 Summary .............................320

9.4 Complements ..........................324

9.5 Problems .............................324

10 Logistic Regression 329

10.1 Binary Regression .......................329

10.2 Binomial Regression ......................335

10.3 Inference .............................340

10.4 Variable Selection .......................350

10.5 Complements ..........................358

10.6 Problems .............................361

11 Poisson Regression 375

11.1 Poisson Regression .......................375

11.2 Inference .............................383

11.3 Variable Selection .......................388

11.4 Complements ..........................393

iii

11.5 Problems .............................395

12 Generalized Linear Models 401

12.1 Introduction ...........................401

12.2 Multiple Linear Regression .................403

12.3 Logistic Regression ......................404

12.4 Poisson Regression .......................406

12.5 Inference and Variable Selection ..............407

12.6 Complements ..........................414

12.7 Problems .............................415

13 Theory for Linear Models 416

13.1 Complements ..........................416

13.2 Problems .............................417

14 Multivariate Models 419

14.1 The Multivariate Normal Distribution ..........420

14.2 Elliptically Contoured Distributions ............424

14.3 Sample Mahalanobis Distances ...............428

14.4 Complements ..........................429

14.5 Problems .............................429

15 1D Regression 433

15.1 Estimating the Suﬃcient Predictor ............436

15.2 Visualizing 1D Regression ..................441

15.3 Predictor Transformations ..................449

15.4 Variable Selection .......................450

15.5 Inference .............................461

15.6 Complements ..........................472

15.7 Problems .............................475

16 Survival Analysis 481

16.1 Univariate Survival Analysis .................482

16.2 Proportional Hazards Regression ..............495

16.2.1 Visualizing the Cox PH Regression Model ....496

16.2.2 Testing and Variable Selection ...........502

16.3 Weibull and Exponential Regression ............509

16.4 Accelerated Failure Time Models .............516

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